Question

The hydrostatic stress for the stress tensor provided below is ......... MPa (in integer). \[ \begin{bmatrix} 150 & 0 & 0 \\ 0 & -100 & 100 \\ 0 & 100 & 250 \end{bmatrix} \, \text{MPa} \]

Hint:

Hydrostatic stress can be calculated by averaging the normal stresses (diagonal elements) of the stress tensor. This represents the isotropic component of the stress state and helps in understanding the material's overall response to stress.

Answer 1

The hydrostatic stress is a measure of the average normal stress in a material. It is calculated as the average of the diagonal components of the stress tensor. In this case, the stress tensor is a 3x3 matrix, where the diagonal elements represent the normal stresses along the \( x \)-, \( y \)-, and \( z \)-axes.

The formula for calculating hydrostatic stress \( \sigma_{{hydrostatic}} \) is:

\[ \sigma_{{hydrostatic}} = \frac{1}{3} (\sigma_{xx} + \sigma_{yy} + \sigma_{zz}) \] Where:
- \( \sigma_{xx} = 150 \, {MPa} \),
- \( \sigma_{yy} = -100 \, {MPa} \),
- \( \sigma_{zz} = 250 \, {MPa} \).

Substituting the values:

\[ \sigma_{{hydrostatic}} = \frac{1}{3} (150 + (-100) + 250) = \frac{1}{3} (300) = 100 \, {MPa} \] Thus, the hydrostatic stress for this stress tensor is \( \mathbf{100 \, {MPa}} \). This result represents the isotropic stress component, which is the same in all directions.